Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support

Abstract

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive support, the so-called foundation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the Signorini condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on arguments of nonlinear equations with multivalued maximal monotone operators and fixed point. Then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the unilateral contact conditions by using an augmented Lagrangian approach. We implement this scheme in a numerical code then we present numerical simulations in the study of two-dimensional test problems, together with various comments and interpretations.Crown

A Quasistatic Electro-Viscoelastic Contact Problem with Adhesion

“…In this work we continue in this line of research, where we extend the result established in [3,20] for contact problem described with the Signorini conditions into contact problem described with the Signorini conditions with adhesion where the obstacle is a perfect insulator and the resistance to tangential motion is generated by the glue, in comparison to which the frictional traction can be neglected. Therefore, the tangential contact traction depends only on the bonding field and the tangential displacement.…”

“…Indeed, (5.20) and (5.21) guarantee that β η (t) ≤ β 0 , and therefore, assumption (3.9) shows that β η (t) ≤ 1 for t ≥ 0, a.e. on 3 . On the other hand, if β η (t 0 ) = 0 at t = t 0 , then it follows from (5.20) and (5.21) thatβ η (t) = 0 for all t ≥ t 0 and therefore, β η (t) = 0 for all t ≥ t 0 , a.e.…”

“…On the other hand, if β η (t 0 ) = 0 at t = t 0 , then it follows from (5.20) and (5.21) thatβ η (t) = 0 for all t ≥ t 0 and therefore, β η (t) = 0 for all t ≥ t 0 , a.e. on 3 . We conclude that 0 ≤ β η (t) ≤ 1 for all t ∈ [0, T ], a.e.…”

“…We conclude that 0 ≤ β η (t) ≤ 1 for all t ∈ [0, T ], a.e. on 3 . Therefore, from the definition of the set Q, we find that β η ∈ Q, which concludes the proof of Lemma 5.3.…”

Analysis of a dynamic electro‐elastic problem

“…Details on the numerical solution of discrete linear piezoelectric problems similar to Problem P hk V can be found in [2][3][4]. We consider an electro-elastic body extended indefinitely in the direction X 1 of a cartesian coordinate system (O, X 1 , X 2 , X 3 ).…”